## 00 - The AttemptHome - - - Physics 2013.epub There is an underlying physical structure to our world that gives rise to a diverse experience of our environment. Physics is based on experiments and relies on intuition to determine the pattern that evolve the universe from the past into the future. This evolution has the hallmarks of Causality and Reversibility. These are two fundamental concepts of Physics which establish the criteria for a Physical Law, otherwise know as a "Law of Nature". A third fundamental concepts is Invariance, which states that wherever and whenever a "Law of Nature" operates the result is always the same. Physics and its language of expression, Mathematics, are human constructions. Physics and Math are both "a work in progress". The "Laws of Nature" do not exist beyond the boundaries of Physics consistent with Causality, Reversibility, and Invariance. The "Laws of Nature" are theoretical models of the Universe. The theories can be incorrect or refined to render a more accurate representation of the universe and can not be proved. The theories can be disproved by experimental evidence. A theory is a tool for making predictions about the environment, as the environment expands the "Law of Nature" changes to accommodate the experimental evidence. Mathematics develops similarly to physical experimentation and theoretical revision. Two fundamental concepts in Mathematics is Consistency and Completeness. There are no contradictions and everything required exists. This requirement is the driving force of evolution in Mathematics. The current state of Mathematics is centuries old and it provides Physics with some impressive Modeling Tools. Mathematics can prove theoretical models of the Mathematics. My favourite number is "e to the i pi". It looks like this, and satisfies the equation . This statement comes from Mathematics free of Physical Constraints yet the elements used in this notation are fundamental to the state of the Universe. The Mathematics can be proved and the interpretation of "e to the i pi" in the "Laws of Nature" is in agreement with experimental Physics. This confluence of thought and experience can lead to an understanding of the nature of Nature. The Understanding of our Universe is Limited by Causality, Reversibility, Invariance, Consistency, Completeness and That's OK. ## 01 - Inertial Reference FrameConsider the existence of a small volume of Space and Time where each tick of time is the same duration of every other tick of time and each increment of space is the same length of every other increment of space. Picture a uniform grid. The Pythagorean theorem states: In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).Moreover, the relationship between the sides of such a triangle, called a right triangle, remains invariant regardless of the triangle's orientation in Space. The criteria of length invariance of an Inertial Reference Frame is a physical property of the Universe. Length can be measured in Physics Experiments. When it conforms to the fundamental concepts of Causality and Reversibility in the framework of Physics and is demonstrated to have Consistency within the scope of Mathematics it can serve as an element in the "Laws of Nature". The relationships discovered in Euclidean Space preserve length invariance. Euclidean Geometry formulate relationships consistent in algebra and number theory explained in geometrical language. Length is measured in Physical Space and is independent of the coordinate system. The coordinate system is constructed of rules that are derived from the properties of Space. The consequences of the Pythagorean theorem supply the rule for distance in Euclidean Space and keep the axes in Cartesian Coordinates perpendicular. The length r is the distance between two points in Physical Space mapped into Euclidean Space and drawn on this page using Cartesian Coordinates.If the two points, p and q, are on a Number Line, which is a One Dimensional Euclidean Space using 1-D Cartesian Coordinates, the line segment pq has a length of r. Coordinate p = (p_{1}) and q = (q_{1}) and If the two points, p and q, lie on a Plane, which is a 2-D Euclidean Space using 2-D Cartesian Coordinates, the line segment pq has a length of r. Coordinate p = (p_{1}, p_{2}) and q = (q_{1}, q_{2}) and If the two points, p and q, are in a Physical Volume, which is a 3-D Euclidean Space using 3-D Cartesian Coordinates, the line segment pq has a length of r. Coordinate p = (p_{1}, p_{2}, p_{3}) and q = (q_{1}, q_{2}, q_{3}) and The pattern of algebra conforms to Length Invariance in Space which is a Real Measurable quantity in Physics and a criteria for an Inertial Reference Frame. Euclidean Space can be extended to higher dimensions and remains consistent. Sigma Σ is a mathematical operator which is used to add up a sequence of expressions that are indexed with a subscript over a range indicated below and above the Sigma symbol. This function to calculate distance is important. It is called aPhysics requires measurable physical quantities and consistent mapping system to analysis the experimental findings in order to build Models which can be Laws of Nature. Inertial Reference Frames are used to verify the universality of proposed Physical Laws. Building on the relationships among the three spatial dimensions in the physical world will lead to a relationship with the forth dimension in the physical world, Time. Justifying a uniform "tick of time" throughout an Inertial Reference Frame required experimentation and evaluation of the Mathematics of Space. There are Geometries, Algebras, and Number Theories which reach a lofty height of abstraction. Fortunately Physics must limit the range of mathematical objects available as tools to build Models of the physical universe within the scope of causality, reversibility, and invariance. A Law of Nature can not produce an unreal result. Physics advances when experimental results support the prediction of a current Law of Nature and when experimental evidence find the prediction of a current Law of Nature wrong. The Laws of Nature within the scope of Physics are subject to change with the results of experiments. Nature is always correct and Physics is a close second. Length is a physical quantity that can be measured in the Domain of Physics. It can be assigned within the Range of Physics. Statements can be made about Length within the scope of Physics. Understanding the Universe happens when two or more lengths are investigated with mathematical tools. Consider the diameter and circumference of a circle. Circles and their 3D relatives, Spheres, exist aplenty in the environment and their diameters with corresponding circumference can be experimentally measured. There is a fixed ratio between circumference and diameter among all circles and spheres. This ratio is so ubiquitous in the universe that it considered a fundamental quality that is incorporated into all the Laws of Nature in a myriad of guises. The name of this ratio is pi, π. Some people will want a numerical value for this number. Its irrational, pi = 3.14159265..., going on forever, like trying to fit a square peg into a round hole. It is a cornerstone relating Space and Time. It is advantageous to see the dynamic property of Space, The three intervals of space and the one interval for time do not change their length in an Inertial Reference Frame. They are in a sense static, of fixed lengths like an assemblage of rigid rods. The dynamic property of Space can be observed as a communication between the elements of space and time. Intuition serves as a guide to understand this communication. The clang of a bell is an event, an object in space with a multitude of forces in concert to produce a clang. The positions of the bell and its clapper can be mapped into an Inertial Reference Frame, but the full pictures requires a mapping of the forces which give rise to the clang. The dynamic property of Space supports the mathematics to render this more complete picture. Anything in the universe that is round or rolls or rotates even if it is not round is governed by the value of pi because space and time have a structure which is fundamentally consistent with pi. In fact, dispersion like any kind of explosion or wave on a pond or sound in the air are constrained by the geometry of space and time where pi is a fundamental quality. Pi is essential to understand the universe because it is an accessible measurable quantity that Physics can lay claim to as being Real. The ninety degree line which equals one-half pi radians appears to be perpendicular to the line labeled zero degrees. And indeed it is. The drawing is a decent representation of the underlying mathematics, algebra and number theory, based on the experimental evidence by measuring circles and spheres abundant in the physical environment. Rather than concentrating on the problem of trying to fit a square peg into a round hole, Physics exploits the triangles which arise from pi. "Soh Cah Toa" is a pleasing phrase to say. It feels good on the face, palate, and in the throat. "Soh Cah Toa" is a mnemonic for the basic trigonometric functions. The trigonometric functions relate a piece of pi to the length of the sides of a triangle. The basic trigonometric functions link three measurements together because pi is fundamental in the Universe. "Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent". My favourite trigonometric function is Cosine and I do not care for Tangent. Just let tangent fade into the background and do its thing with Calculus. Cosine wraps around a circle along with Sine. Space is independent of a coordinate system. Physics requires a measurement. Pick two points in Space, call one point p and q for the other. The distance from p to q is the length r of the line segment pq that connects them. The Cartesian Coordinate System can be used to map the measurement by specifying ordered-pairs (x, y) for each of the points p and q. The mathematics that support a Coordinate System ensure that the Physical Space is unaffected by the coordinate system. To be useful for physical analysis the coordinate system must embody as many properties of Space that it can. Cartesian Coordinates are such a useful tool as are Polar Coordinates. Algebra and Trigonometry are consistent with each other. The Polar Coordinate System can be used to map the measurement by specifying ordered-pairs (x, y) for each of the points p and q.Some physical quantities that appear as a confusing mess in algebra can be expressed cleanly and understandable in trigonometry. It is not simply a matter of convenience and legibility, it highlights underlying principles functioning in the universe. A cosine can be used as a first approximation to model a wave on the surface of a pond which gives a hint of motion and alludes to the passage of time even in a static drawing. The Mathematics used to construct an Inertial Reference Frame that support the concept of an invariant Length in Space points to an invariant tick in Time. There is a small amount of mathematical machinery that address this "pointing business" which deserves mentioning even without a proper depth of explanation. When Algebra was young it presented a problem that mathematicians could not solve with Real Numbers. That problem was x squared equals negative one. Actually, it was that the square root of any negative Real Number is not included in the set of all Real Numbers. Thus it was discovered that Imaginary Numbers need to exist. They are as real as Real Numbers but they are saddled with the name Imaginary. Imaginary Numbers have the same properties as Real Numbers extended by the consequences that led to their discovery.The lower case letter i denotes the imaginary number. More precisely i is the imaginary unit or unit imaginary number. It acts like 1 for addition, subtraction, multiplication, and division but it carries a direction with it. This is not so strange when considering that positive one and negative one are in opposite directions from zero. The set of Imaginary Numbers are perpendicular to the set of Real Numbers intersecting at the common numerical value zero. The union of these two sets of Numbers form the set of Complex Numbers. If the Real Numbers are laid out on a line running left to right, the Imaginary Numbers would lie on a line running up and down. This looks like Cartesian Coordinates, but the x-axis are Real Numbers and the y-axis is Imaginary Numbers.There is an interesting property of i that make Complex Numbers useful and point to a fundamental unity in Nature. Change happens. Physics endeavours to explain what changes and how it changes. There is a number that is consistent with the Mathematics mentioned thus far that opens a vast vista of tools for Physics. Euler's Number, e, is found in architecture, electrical engineering, hydrodynamics, and a myriad of technologies above and below these in sophistication. At the core of these technologies is a topic in Physics called Mechanics. Most of Mechanics is on the scale of the size of a human and observed with few or no assistance devices. Even at this narrow resolution humans have discovered a number of fundamental features of the Universe. The concept of an invariant length and invariant time is very old. It was reasoned by the Ancients that the World existed in an Absolute Space and and Absolute Time. Space and Time was an immutable Stage on which the People played out their lives. The elements of the Natural World followed Rules that could be discovered at first solely by Reason then later, much later, by Experimentation and Scientific Method. The tract thus far has been largely mathematical while paying deference to the conceptual foundation of Physics: causality, reversibility, and invariance. The theory of Absolute Space and Absolute Time held authority for centuries and it is ill advised to dismiss it outright. Mathematics reveal insights into this theory which led to its refinement as the concept of an Inertial Reference Frame. Simply put an Inertial Reference Frame is a neighbourhood within the Universe where Space and Time are uniform and consistent. Home |